# A Universality Law For Sign Correlations of Eigenfunctions of Differential Operators

## TL;DR

This paper proves a universal law describing the frequency of sign correlations between eigenfunctions of differential operators, showing that the proportion of same-sign pairs converges to a value between 1/3 and 2/3, which is optimal.

## Contribution

It establishes a universal sign correlation law for eigenfunctions satisfying WKB approximation, including classical orthogonal polynomials and Schrödinger eigenfunctions, with proven bounds.

## Key findings

- The proportion of same-sign pairs converges to a limit between 1/3 and 2/3.
- The bounds of 1/3 and 2/3 are proven to be optimal.
- The results extend to other similar questions and open problems are discussed.

## Abstract

We establish a universality law for sequences of functions $\{w_n\}_{n \in \mathbb{N}}$ satisfying a form of WKB approximation on compact intervals. This includes eigenfunctions of generic Schr\"odinger operators, as well as Laguerre and Chebyshev polynomials. Given two distinct points $x, y \in \mathbb{R}$, we ask how often do $w_n(x)$ and $w_n(y)$ have the same sign. Asymptotically, one would expect this to be true half the time, but this turns out to not always be the case. Under certain natural assumptions, we prove that, for all $x \neq y$, $$ \frac{1}{3} \leq \lim_{N \to \infty} \frac{1}{N} \# \left\{0 \leq n < N: \mathrm{sgn}(w_n(x)) = \mathrm{sgn}(w_n(y)) \right\} \leq \frac{2}{3}, $$ and that these bounds are optimal, and can be attained. Our methods extend to other questions of similar flavor and we also discuss a number of open problems.